Two things (which don't address the root cause)
1) if you don't have the fitted OEM foam block that fits inside the shifter hole - just get a cheap foam car-washing sponge and cut a small hole in it, then stuff that down into the gap between the trans and body. Less heat, less noise.
2) I made the T5 in my wagon (not the same thing at all, of course) quite noticeably quieter by changing the shift knob from a solid hard plastic round knob to a leather-wrapped knob with a little sound deadening inside. The stick vibrates, but the knob is one of the ways that vibration gets turned into noise inside the car.